The same questions can be considered more generally in geometrically finite hyperbolic manifolds. The questions mentioned in Section REF  on the induced metrics and third fundamental forms on the boundary of geodesically convex subsets of quasifuchsian manifolds can be stated, basically in the same manner, for Kleinian manifolds. For smooth metrics of curvature \(K>-1\)  (resp. third fundamental forms of curvature \(K<1\)  with closed, contractible geodesics of length \(L>2\pi \) ) the results of [1]}, [2]} also apply for convex co-compact hyperbolic manifolds. For pleated surfaces (induced metrics of constant curvature \(-1\) , resp. measured bending laminations) Conjectures REF  and REF  have been stated by Thurston in this setting, and the existence parts also hold for both conjectures.
